But both these weights, the one going upon the short journey, and the other upon the long, will start down exactly the same inclination or declivity. The reader will see that this must be the case if he will draw two circles on paper round a common centre, the one at the distance of one inch, and the other at the distance of nine inches. Having done this, let him cut a notch out of the paper, extending through both the circles to the centre, and including a twelfth part, or thirty degrees, of each between its converging sides. He will then observe, that the two arcs cut out by the notch are everywhere concentric with each other; therefore, their beginnings and endings are concentric or inclined in exactly the same degree to a perpendicular crossing their centres. These concentric beginnings and endings represent correctly the concentric directions in which the swinging weights commence their downward movements.
Now, since it has been shewn that bodies begin to run down equal descents with equal velocities, it follows that the weight on the short string and that on the long string must commence to move down the concentric curves of their respective arcs at an equal rate. But it has been also shewn that the one of these weights has a nine times longer journey to perform than the other; it is clear, therefore, that both cannot accomplish their respective distances in the same time. The weight on the shortest string in reality makes three vibrations, and the weight on the string that is next to this in length makes two vibrations, while the weight on the longest string is occupied about one; and the differences would be as 9, 4, and 1, instead of as 3, 2, 1, but that the weights moving in the longer arcs benefit most from acceleration of velocity. Although all the vibrating bodies begin to move at equal rates, they pass the central positions directly beneath their points of suspension at unequal ones. Those that have been the longest in getting down to these positions, have of necessity increased their paces the most while upon their route.
Suspended weights, then, only vibrate in equal times when hung upon equal strings; but they continue to make vibrations in equal times notwithstanding the diminution of the arcs in which they swing. This was the fact that caught the attention of Galileo; he observed that the vibrations of the lamp slowly died away as the effect of the disturbing force was destroyed bit by bit, but that, nevertheless, the last faint vibration that caught his eye, took the same apparent time for its performance as the fullest and longest one in the series.